Let's continue to study fractions. Today we will talk about their comparison. The topic is interesting and useful. It will allow a beginner to feel like a scientist in a white coat.
The essence of comparing fractions is to find out which of two fractions is greater or less.
To answer the question which of two fractions is greater or less, use such as more (>) or less (<).
Mathematicians have already taken care of ready-made rules that allow them to immediately answer the question of which fraction is larger and which is smaller. These rules can be safely applied.
We will look at all these rules and try to figure out why this happens.
Lesson contentComparing fractions with the same denominators
The fractions that need to be compared are different. The best case is when the fractions have the same denominators, but different numerators. In this case, the following rule applies:
Of two fractions with the same denominator, the fraction with the larger numerator is greater. And accordingly, the fraction with the smaller numerator will be smaller.
For example, let's compare fractions and answer which of these fractions is larger. Here the denominators are the same, but the numerators are different. The fraction has a greater numerator than the fraction. This means the fraction is greater than . That's how we answer. You must answer using the more icon (>)
This example can be easily understood if we remember about pizzas, which are divided into four parts. There are more pizzas than pizzas:
Comparing fractions with the same numerators
The next case we can get into is when the numerators of the fractions are the same, but the denominators are different. For such cases, the following rule is provided:
Of two fractions with the same numerators, the fraction with the smaller denominator is greater. And accordingly, the fraction whose denominator is larger is smaller.
For example, let's compare the fractions and . These fractions have the same numerators. A fraction has a smaller denominator than a fraction. This means that the fraction is greater than the fraction. So we answer:
This example can be easily understood if we remember about pizzas, which are divided into three and four parts. There are more pizzas than pizzas:
Everyone will agree that the first pizza is bigger than the second.
Comparing fractions with different numerators and different denominators
It often happens that you have to compare fractions with different numerators and different denominators.
For example, compare fractions and . To answer the question which of these fractions is greater or less, you need to bring them to the same (common) denominator. Then you can easily determine which fraction is greater or less.
Let's bring the fractions to the same (common) denominator. Let's find the LCM of the denominators of both fractions. LCM of the denominators of the fractions and this is the number 6.
Now we find additional factors for each fraction. Let's divide the LCM by the denominator of the first fraction. LCM is the number 6, and the denominator of the first fraction is the number 2. Divide 6 by 2, we get an additional factor of 3. We write it above the first fraction:
Now let's find the second additional factor. Divide the LCM by the denominator of the second fraction. LCM is the number 6, and the denominator of the second fraction is the number 3. Divide 6 by 3, we get an additional factor of 2. We write it above the second fraction:
Let's multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to compare such fractions. Of two fractions with the same denominator, the fraction with the larger numerator is greater:
The rule is the rule, and we will try to figure out why it is more than . To do this, select the whole part in the fraction. There is no need to highlight anything in the fraction, since the fraction is already proper.
After isolating the integer part in the fraction, we obtain the following expression:
Now you can easily understand why more than . Let's draw these fractions as pizzas:
2 whole pizzas and pizzas, more than pizzas.
Subtraction of mixed numbers. Difficult cases.
When subtracting mixed numbers, you can sometimes find that things aren't going as smoothly as you'd like. It often happens that when solving an example, the answer is not what it should be.
When subtracting numbers, the minuend must be greater than the subtrahend. Only in this case will a normal answer be received.
For example, 10−8=2
10 - decrementable
8 - subtrahend
2 - difference
The minuend 10 is greater than the subtrahend 8, so we get the normal answer 2.
Now let's see what happens if the minuend is less than the subtrahend. Example 5−7=−2
5—decreasable
7 - subtrahend
−2 — difference
In this case, we go beyond the limits of the numbers we are accustomed to and find ourselves in the world of negative numbers, where it is too early for us to walk, and even dangerous. To work with negative numbers, you need an appropriate math training, which we haven't received yet.
If, when solving subtraction examples, you find that the minuend is less than the subtrahend, then you can skip this example for now. It is permissible to work with negative numbers only after studying them.
The situation is the same with fractions. The minuend must be greater than the subtrahend. Only in this case will it be possible to get a normal answer. And to understand whether the fraction being reduced is greater than the fraction being subtracted, you need to be able to compare these fractions.
For example, let's solve the example.
This is an example of subtraction. To solve it, you need to check whether the fraction being reduced is greater than the fraction being subtracted. more than
so we can safely return to the example and solve it:
Now let's solve this example
We check whether the fraction being reduced is greater than the fraction being subtracted. We find that it is less:
In this case, it is wiser to stop and not continue further calculation. Let's return to this example when we study negative numbers.
It is also advisable to check mixed numbers before subtraction. For example, let's find the value of the expression .
First, let's check whether the mixed number being reduced is greater than the mixed number being subtracted. To do this, we convert mixed numbers to improper fractions:
We received fractions with different numerators and different denominators. To compare such fractions, you need to bring them to the same (common) denominator. We will not describe in detail how to do this. If you have difficulty, be sure to repeat.
After reducing the fractions to the same denominator, we obtain the following expression:
Now you need to compare the fractions and . These are fractions with the same denominators. Of two fractions with the same denominator, the fraction with the larger numerator is greater.
The fraction has a greater numerator than the fraction. This means that the fraction is greater than the fraction.
This means that the minuend is greater than the subtrahend
This means we can return to our example and safely solve it:
Example 3. Find the value of an expression
Let's check whether the minuend is greater than the subtrahend.
Let's convert mixed numbers to improper fractions:
We received fractions with different numerators and different denominators. Let us reduce these fractions to the same (common) denominator:
Now let's compare the fractions and . A fraction has a numerator less than a fraction, which means the fraction is less than a fraction
Not only can prime numbers be compared, but fractions too. After all, a fraction is the same number as, for example, natural numbers. You only need to know the rules by which fractions are compared.
Comparing fractions with the same denominators.
If two fractions have the same denominators, then it is easy to compare such fractions.
To compare fractions with the same denominators, you need to compare their numerators. The fraction that has a larger numerator is larger.
Let's look at an example:
Compare the fractions \(\frac(7)(26)\) and \(\frac(13)(26)\).
The denominators of both fractions are the same and equal to 26, so we compare the numerators. The number 13 is greater than 7. We get:
\(\frac(7)(26)< \frac{13}{26}\)
Comparing fractions with equal numerators.
If a fraction has the same numerators, then the fraction with the smaller denominator is greater.
This rule can be understood by giving an example from life. We have cake. 5 or 11 guests can come to visit us. If 5 guests come, then we will cut the cake into 5 equal pieces, and if 11 guests come, then we will divide it into 11 equal pieces. Now think about the situation in which one guest will have a piece of cake larger size? Of course, when 5 guests arrive, there will be a larger piece of cake.
Or another example. We have 20 candies. We can give the candy equally to 4 friends or divide the candy equally among 10 friends. In what case will each friend have more candies? Of course, when we share with only 4 friends, the number of candies for each friend will be greater. Let's check this problem mathematically.
\(\frac(20)(4) > \frac(20)(10)\)
If we solve these fractions before, we get the numbers \(\frac(20)(4) = 5\) and \(\frac(20)(10) = 2\). We get that 5 > 2
This is the rule for comparing fractions with the same numerators.
Let's look at another example.
Compare fractions with the same numerator \(\frac(1)(17)\) and \(\frac(1)(15)\) .
Since the numerators are the same, the fraction with the smaller denominator is larger.
\(\frac(1)(17)< \frac{1}{15}\)
Comparing fractions with different denominators and numerators.
To compare fractions with different denominators, you need to reduce the fractions to , and then compare the numerators.
Compare the fractions \(\frac(2)(3)\) and \(\frac(5)(7)\).
First, let's find the common denominator of the fractions. It will be equal to the number 21.
\(\begin(align)&\frac(2)(3) = \frac(2 \times 7)(3 \times 7) = \frac(14)(21)\\\\&\frac(5) (7) = \frac(5 \times 3)(7 \times 3) = \frac(15)(21)\\\\ \end(align)\)
Then we move on to comparing numerators. Rule for comparing fractions with the same denominators.
\(\begin(align)&\frac(14)(21)< \frac{15}{21}\\\\&\frac{2}{3} < \frac{5}{7}\\\\ \end{align}\)
Comparison.
An improper fraction is always larger than a proper fraction. Because an improper fraction is greater than 1, and a proper fraction is less than 1.
Example:
Compare the fractions \(\frac(11)(13)\) and \(\frac(8)(7)\).
The fraction \(\frac(8)(7)\) is improper and is greater than 1.
\(1 < \frac{8}{7}\)
The fraction \(\frac(11)(13)\) is correct and it is less than 1. Let’s compare:
\(1 > \frac(11)(13)\)
We get, \(\frac(11)(13)< \frac{8}{7}\)
Related questions:
How to compare fractions with different denominators?
Answer: you need to bring the fractions to a common denominator and then compare their numerators.
How to compare fractions?
Answer: First you need to decide what category fractions belong to: they have a common denominator, they have a common numerator, they do not have a common denominator and numerator, or you have a proper and improper fraction. After classifying fractions, apply the appropriate comparison rule.
What is comparing fractions with the same numerators?
Answer: If fractions have the same numerators, the fraction with the smaller denominator is larger.
Example #1:
Compare the fractions \(\frac(11)(12)\) and \(\frac(13)(16)\).
Solution:
Since there are no identical numerators or denominators, we apply the rule of comparison with different denominators. We need to find a common denominator. The common denominator will be 96. Let's reduce the fractions to a common denominator. Multiply the first fraction \(\frac(11)(12)\) by an additional factor of 8, and multiply the second fraction \(\frac(13)(16)\) by 6.
\(\begin(align)&\frac(11)(12) = \frac(11 \times 8)(12 \times 8) = \frac(88)(96)\\\\&\frac(13) (16) = \frac(13 \times 6)(16 \times 6) = \frac(78)(96)\\\\ \end(align)\)
We compare fractions with numerators, the fraction with the larger numerator is larger.
\(\begin(align)&\frac(88)(96) > \frac(78)(96)\\\\&\frac(11)(12) > \frac(13)(16)\\\ \\end(align)\)
Example #2:
Compare a proper fraction to one?
Solution:
Any proper fraction is always less than 1.
Task #1:
The son and father were playing football. The son hit the goal 5 times out of 10 approaches. And dad hit the goal 3 times out of 5 approaches. Whose result is better?
Solution:
The son hit 5 times out of 10 possible approaches. Let's write it as a fraction \(\frac(5)(10)\).
Dad hit 3 times out of 5 possible approaches. Let's write it as a fraction \(\frac(3)(5)\).
Let's compare fractions. We have different numerators and denominators, let's reduce them to one denominator. The common denominator will be 10.
\(\begin(align)&\frac(3)(5) = \frac(3 \times 2)(5 \times 2) = \frac(6)(10)\\\\&\frac(5) (10)< \frac{6}{10}\\\\&\frac{5}{10} < \frac{3}{5}\\\\ \end{align}\)
Answer: Dad has a better result.
The rules for comparing ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the denominators (same or different) of the fractions being compared.
This section discusses options for comparing fractions that have the same numerators or denominators.
Rule. To compare two fractions with the same denominators, you need to compare their numerators. Greater (less) is a fraction whose numerator is greater (less).
For example, compare fractions:
Rule. To compare proper fractions with like numerators, you need to compare their denominators. Greater (less) is a fraction whose denominator is less (greater).
For example, compare fractions: 
Comparing proper, improper and mixed fractions with each other
Rule. Improper and mixed fractions are always larger than any proper fraction.
A proper fraction is by definition less than 1, so improper and mixed fractions (those containing a number equal to or greater than 1) are greater than a proper fraction.
Rule. Of two mixed fractions, the one whose whole part of the fraction is greater (less) is greater (smaller). When the whole parts of mixed fractions are equal, the fraction with the larger (smaller) fractional part is greater (smaller).
The rules for comparing ordinary fractions depend on the type of fraction (proper, improper, mixed fraction) and on the denominators (same or different) of the fractions being compared. Rule. To compare two fractions with the same denominators, you need to compare their numerators. Greater (less) is a fraction whose numerator is greater (less). For example, compare fractions:
Comparing proper, improper and mixed fractions with each other.
Rule. Improper and mixed fractions are always larger than any proper fraction. A proper fraction is by definition less than 1, so improper and mixed fractions (those containing a number equal to or greater than 1) are greater than a proper fraction.
Rule. Of two mixed fractions, the one whose whole part of the fraction is greater (less) is greater (smaller). When the whole parts of mixed fractions are equal, the fraction with the larger (smaller) fractional part is greater (smaller).
For example, compare fractions:
Similar to comparing natural numbers on the number line, the larger fraction is to the right of the smaller fraction.
IN everyday life We often have to compare fractional quantities. Most often this does not cause any difficulties. Indeed, everyone understands that half an apple is larger than a quarter. But when it comes to writing it down as a mathematical expression, it can get confusing. Using the following mathematical rules, you can easily cope with this task.
How to compare fractions with the same denominators
Such fractions are most convenient to compare. In this case, use the rule:
Of two fractions with the same denominators but different numerators, the larger is the one whose numerator is larger, and the smaller is the one whose numerator is smaller.
For example, compare the fractions 3/8 and 5/8. The denominators in this example are equal, so we apply this rule. 3<5 и 3/8 меньше, чем 5/8.
Indeed, if you cut two pizzas into 8 slices, then 3/8 of a slice is always less than 5/8.
Comparing fractions with like numerators and unlike denominators
In this case, the sizes of the denominator shares are compared. The rule to be applied is:
If two fractions have equal numerators, then the fraction whose denominator is smaller is greater.
For example, compare the fractions 3/4 and 3/8. In this example, the numerators are equal, which means we use the second rule. The fraction 3/4 has a smaller denominator than the fraction 3/8. Therefore 3/4>3/8
Indeed, if you eat 3 slices of pizza divided into 4 parts, you will be more full than if you ate 3 slices of pizza divided into 8 parts.
Comparing fractions with different numerators and denominators
Let's apply the third rule:
Comparing fractions with different denominators should lead to comparing fractions with the same denominators. To do this, you need to reduce the fractions to a common denominator and use the first rule.
For example, you need to compare fractions and . To determine the larger fraction, we reduce these two fractions to a common denominator:
- Now let's find the second additional factor: 6:3=2. We write it above the second fraction:
